Force times distance (if the force is constant)

[speeding electron]

The idea of energy is, it could be argued, intuitive. If a object is moving, or if it is hot, then it is natural to think of some sort vibrancy, some sort of rousing inner property of the object. As I understand it is defined by: force times distance (if the force is constant). Who discovered that is was this that was conserved, i.e. that it was this product that gave the fundamental property of energy?

 

[Gokul43201]

F.d is simply the work done on a body. It may lead to a change in energy, but is itself not a measure of energy. So there is no meaning to the statement "F.d is conserved."

 

[speeding electron]

Yes, but why does this particular product equal the energy change?

 

[JohnDubYa]

This is a good opportunity to nail down precisely the definition of energy. I'll start:

We deemed the product of the force and distance an important physical quantity (it in essence tells you how much you accomplished in a day's work). We then defined energy as the ability to produce this amount of Fd. Later it was established that this energy was a conserved quantity. Noether's theorem later proved it.

Are there problems with the above description? I think so, but at this time I am not sure where. Anyone?

 

[arildno]

Let us review Newton's 2.law for an object with constant mass.
This states:
$$\vec{F}=m\vec{a}$$
Multiply this equation with the velocity:
$$\vec{F}\cdot\vec{v}=m\vec{a}\cdot\vec{v}$$

Now, we may rewrite the right-hand side:
$$\vec{F}\cdot\vec{v}=\frac{d}{dt}(\frac{1}{2}m\vec{v}^{2})$$

This equation is rather instructive; it says that the rate of change of an object's kinetic energy equals the net power from the external forces acting upon the object.
We integrate in time, and get:
$$\int_{0}^{t}\vec{F}\cdot\frac{d\vec{x}}{d\tau}d\tau=\bigtriangleup(\mathcal{K}_{E})$$

Recognizing $$\frac{d\vec{x}}{d\tau}d\tau$$ as the infinitesemal distance $$d\vec{x}$$ covered in time interval $$d\tau$$ we finally get the expression:
$$\bigtriangleup(\mathcal{K}_{E})=\oint\vec{F}\cdot{d\vec{x}}$$

which is the formula relating the change in kinetic energy to the work done along the object's path.

 

[JohnDubYa]

Which is the work-energy theorem for kinetic energy. But the question was asked about energy, not just kinetic energy.

The work-energy theorem for potential energy states that the work done by conservative forces is given by the negative change in potential energy. Also, the work done by nonconservative forces is given by the change in total (mechanical) energy of the body. So that has to be included in the definition of energy.

 

[Gokul43201]

Yes, but why does this particular product equal the energy change?

It has dimensions of energy.

 

[jdavel]

Gokul said, "It has dimensions of energy."

So does torque.

 

[TALewis]

Gokul said, "It has dimensions of energy."

So does torque.

Good catch.

 

[speeding electron]

So it was shown that energy is conserved after we defined it as equal to Fd? Does that mean that it was just a lucky guess? What was the use of assigning significance to this value before we knew it was conserved?

 

[JohnDubYa]

To lift a box you must apply a force through a distance. Lifting a box to a higher height is an accomplishment, so F X d is an important property.

But looking back, my explanation sucks.

 

[arildno]

"Energy can neither be created nor destoyed.."
I thought that was an axiom modern physics adheres to(?)

 

[speeding electron]

JohnDubYa: I take your point, and that's fair enough. My original point was that in a way they struck it lucky, because for all they knew before the discovery it was conserved, it could have been that in nature what is conserved is Forces times the square of the distance, or something else. Did they suspect it was conserved (Fd that is) when they assigned it a significance?

 

[JohnDubYa]

To be honest, I'm not sure how the concept of work was originally fabricated. Good question.

 

[JohnDubYa]

By the way, answering your question would be a good technical paper for an undergraduate course.

 

[TALewis]

You might be interested in looking up Joule's experiment. Joule used an apparatus that included a mass that was strung around a paddle, so that the paddle stirred a bucket of water as the mass fell. In this experiment, Joule related the work gravity does on a falling mass (weight*distance, or mgh) to the increase in the water's temperature due to the spinning paddle. His result is known historically as "the mechanical equivalent of heat," showing the relationship between mechanical energy and internal energy.

 

[Gokul43201]

Gokul said, "It has dimensions of energy."

So does torque.

Okay, let me rephrase. It is a scalar quantity with dimensions of energy.

In other words, find me any scalar with dimensions of energy, and there is some way to see that quantity as being the change in energy of a system. F.d is just one such scalar.

How about (dF/dl).A ? That too has dimensions of energy, doesn't it ?

Yes, and it is the surface energy of a liquid, with surface tension= F/l and surface area = A.

 

[Gokul43201]

JohnDubYa: My original point was that in a way they struck it lucky, because for all they knew before the discovery it was conserved, it could have been that in nature what is conserved is Forces times the square of the distance, or something else. Did they suspect it was conserved (Fd that is) when they assigned it a significance?

Please ! F.d is NOT conserved. In fact, while gravity (for example) is a conservative force (integral over a closed path vanishes) this is not true for a general F.dx.

There is nothing special about F.d. It is just the right way to calculate the change in energy of certain systems. Energy is conserved, not because of F.d. Energy would be conserved, no matter how it was calculated.

 

[jdavel]

speeding electron asked, "Did they suspect it was conserved (Fd that is) when they assigned it a significance?"

You've nailed it!

Energy is significant in physics for exactly one reason: it's conserved.

 

[Gokul43201]

Take a horizontal tube of varying cross section, with pistons at the ends, filled with an incompressible fluid. Pascal found that, anywhere along this tube, F/A is conserved. This is the principle on which hydraulic brakes work.

 

[speeding electron]

Thank you for the informative replies. So they just began to realize that work done equaled the work done on the sytem? What I'm trying to get at here is this: had they already defined the work done as F.dx, and then they realized that that equaled the energy change, or was it that they had defined the work done as the energy change, and then they realized that the work done was F.dx?
I suspect TALewis answered this by citing Joule's experiment. I'll go and look at that now.